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1. Local topological order and boundary algebras

   https://doi.org/10.1017/fms.2025.16  

   Jones, Corey; Naaijkens, Pieter; Penneys, David; Wallick, Daniel; Izumi, Masaki (January 2025, Forum of Mathematics, Sigma)

   Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on$$\mathbb {Z}^{k}$$, we define a local net of boundary algebras on$$\mathbb {Z}^{k-1}$$, which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type$$\mathrm {II}$$, and we show that Levin-Wen models can have cone algebras of type$$\mathrm {III}$$. Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states. 

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2. Ground state degeneracy on torus in a family of ZN toric code

   https://doi.org/10.1063/5.0134010  

   Watanabe, Haruki; Cheng, Meng; Fuji, Yohei (May 2023, Journal of Mathematical Physics)

   Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species. 

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3. Levin-Wen is a Gauge Theory: Entanglement from Topology

   https://doi.org/10.1007/s00220-024-05144-x  

   Kawagoe, Kyle; Jones, Corey; Sanford, Sean; Green, David; Penneys, David (October 2024, Communications in Mathematical Physics)

   Abstract We show that the Levin-Wen model of a unitary fusion category$${\mathcal {C}}$$ $C$is a gauge theory with gauge symmetry given by the tube algebra$${\text {Tube}}({\mathcal {C}})$$ $\text{Tube}\left(C\right)$. In particular, we define a model corresponding to a$${\text {Tube}}({\mathcal {C}})$$ $\text{Tube}\left(C\right)$symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case$${\mathcal {C}}=\textsf{Hilb}(G,\omega )$$ $C=\mathrm{Hilb}(G,\omega )$, we show how our procedure reduces to the twisted gauging of a trivalG-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibbonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter. 

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4. Entanglement entropy of a color flux tube in (2+1)D Yang-Mills theory

   https://doi.org/10.1007/JHEP12(2024)177  

   Amorosso, Rocco; Syritsyn, Sergey; Venugopalan, Raju (December 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We construct a novel flux tube entanglement entropy (FTE²), defined as the excess entanglement entropy relative to the vacuum of a region of color flux stretching between a heavy quark-anti-quark pair in pure gauge Yang-Mills theory. We show that FTE²can be expressed in terms of correlators of Polyakov loops, is manifestly gauge-invariant, and therefore free of the ambiguities in computations of the entanglement entropy in gauge theories related to the choice of the center algebra. Employing the replica trick, we compute FTE²for SU(2) Yang-Mills theory in (2+1)D and demonstrate that it is finite in the continuum limit. We explore the properties of FTE²for a half-slab geometry, which allows us to vary the width and location of the slab, and the extent to which the slab cross-cuts the color flux tube. Following the intuition provided by computations of FTE²in (1+1)D, and in a thin string model, we examine the extent to which our FTE²results can be interpreted as the sum of an internal color entropy and a vibrational entropy corresponding to the transverse excitations of the string. 

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5. Dihedral twist liquid models from emergent Majorana fermions

   https://doi.org/10.22331/q-2023-03-30-967  

   Teo, Jeffrey C.; Hu, Yichen (March 2023, Quantum)

   We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the S O ( N ) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral D k symmetry of S O ( 2 n ) 1 , and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C / G twist liquids, where G = Z 2 for C = U ( 1 ) l , S U ( n ) 1 , and G = Z 2 , Z k , D k for C = S O ( 2 n ) 1 . We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in S O ( 2 n ) 1 / G by identifying the non-chiral components of the twist liquids with discrete gauge theories. 

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